In this paper we relate the Fefferman–Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tractor bundle and the normal standard tractor connection, which are equivalent to...

Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to ...

We present an explicit numerical method to obtain the Cartan-KhanejaGlaser decomposition of a general element G ∈ SU(2 ) in terms of its ‘Cartan’ and ‘non-Cartan’ components. This effectively factors G in terms of group elements that belong in SU(2) with n < N , a procedure that an be iterated down to n = 2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained b...

Abstract In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space $${\mathbb {R}}^3$$ R 3 to case Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., Riemannian 3-manifolds whose metr...

Standard notation and terminology of differential geometry and general relativity are used in this article. All considerations are local so that the four-dimensional space-time M is assumed to be a smooth manifold diffeomorphic to R4. It is endowed with a metric tensor g of signature (1, 3) and a linear connection defining the covariant differentiation of tensor fields. Greek indices range from...